Correct, both the Convox ECS rack and Empire are AWS exclusive. Convox has a public-beta of their rack running on Kubernetes which is cloud agnostic as it only needs a Kubernetes cluster to operate in.
Does Convox provision and manage the underlying Kubernetes cluster or depends on the Cloud provider's managed Kubernetes offering?
Convox does not manage the underlying Kubernetes cluster. If you're using EKS it's managed by AWS, GKE by GCP, AKS by Azure, etc. If you're using a self-hosted cluster, then it's up to you to manage it. The Convox platform runs as Kubernetes CRDS so it is cloud agnostic.